This paper considers non-convex mixed-integer nonlinear programming where nonlinearity comes in the presence of the two-dimensional euclidean norm in the objective or the constraints. We build from the euclidean norm piecewise linearization proposed by [Camino et al., 2019], that allows to solve such non-convex problems via mixed-integer linear programming with an arbitrary approximation guarantee. Theoretical results that make this linearization able to satisfy any given approximation level with the minimum number of pieces are established. An extension of the piecewise linearization approach sharing the same theoretical properties is proposed for elliptic constraints and/or objective. An application of the elliptic linearization to a non-convex beam layout mixed optimization problem coming from an industrial application shows the practical appeal of the approach.